The Gamma distribution is a continuous probability distribution defined on the positive real line $ (0,\infty) $. It is commonly used to model waiting times, lifetimes of components, and various stochastic processes. The distribution is characterized by two parameters: a shape parameter $k>0$ (also denoted $\alpha$ or $r$) and a scale parameter $\theta>0$ (also denoted $\beta$ or $1/\lambda$). Its probability density function (PDF) is
$$ f(x; k, \theta)=\frac{x^{k-1} e^{-x/\theta}}{\Gamma(k),\theta^{k}},\qquad x>0, $$
where $\Gamma(k)=\int_{0}^{\infty} t^{k-1}e^{-t},dt$ is the gamma function, which generalizes the factorial function to non‑integer arguments.
Alternative Parameterizations
- Rate parameterization: Using a rate $\lambda = 1/\theta$, the PDF becomes
$$ f(x; k, \lambda)=\frac{\lambda^{k} x^{k-1} e^{-\lambda x}}{\Gamma(k)}. $$ - Shape–scale and shape–rate forms are interchangeable via $\theta = 1/\lambda$.
Moments
- Mean: $\mathbb{E}[X]=k\theta$.
- Variance: $\operatorname{Var}(X)=k\theta^{2}$.
- Higher‑order moments: For $n\in\mathbb{N}$, $\mathbb{E}[X^{n}] = \theta^{n},\frac{\Gamma(k+n)}{\Gamma(k)}$.
Relationship to Other Distributions
- When $k=1$, the Gamma distribution reduces to the exponential distribution with rate $\lambda$.
- For integer $k=n$, the distribution is known as the Erlang distribution, frequently used in queuing theory.
- The chi‑squared distribution with $ u$ degrees of freedom is a special case: $\chi^{2}_{ u} \sim \Gamma(k= u/2,\theta=2)$.
- Sums of independent Gamma‑distributed variables with a common scale (or rate) parameter are again Gamma‑distributed, with shape parameters adding.
Generation of Random Variates
Standard algorithms include:
- Ahrens–Dieter and Marsaglia–Tsang methods for shape $k\ge 1$.
- Acceptance‑rejection or transformed‑rejection techniques for $0<k<1$.
Parameter Estimation
- Method of moments: equate sample mean $\bar{x}$ and sample variance $s^{2}$ to the theoretical mean and variance, yielding $\hat{k} = \frac{\bar{x}^{2}}{s^{2}}$ and $\hat{\theta}= \frac{s^{2}}{\bar{x}}$.
- Maximum likelihood estimation (MLE): the log‑likelihood
$$ \ell(k,\theta)= (k-1)\sum_{i}\ln x_i -\frac{1}{\theta}\sum_{i}x_i - n\bigl[ \ln\Gamma(k)+k\ln\theta \bigr] $$ leads to coupled equations that are solved iteratively (e.g., Newton–Raphson).
Applications
- Reliability engineering: modeling failure times of devices with a “wear‑out” phase.
- Hydrology: describing rainfall amounts and river discharge.
- Bayesian statistics: serving as a conjugate prior for the rate parameter of a Poisson distribution and for the precision (inverse variance) of a normal distribution.
- Queueing theory: inter‑arrival times and service times in various queuing models.
- Finance: modeling aggregated insurance claim sizes and stochastic volatility.
Historical Note
The gamma function, introduced by Leonhard Euler in the 18th century, underlies the distribution’s name. The Gamma distribution itself was formally described in the early 20th century within the context of chi‑squared and Erlang models; the term “Gamma” reflects its connection to the gamma function rather than a specific individual.
Mathematical Properties
- Closure under scaling: If $X\sim\Gamma(k,\theta)$ and $c>0$, then $cX\sim\Gamma(k,c\theta)$.
- Infinite divisibility: The Gamma distribution is infinitely divisible; it can be represented as a sum of an arbitrary number of independent identically distributed Gamma variables with appropriately adjusted shape parameters.
- Moment‑generating function: $M_X(t)= (1-\theta t)^{-k}$ for $t<1/\theta$.
- Characteristic function: $\phi_X(t)= (1-i\theta t)^{-k}$.
The Gamma distribution remains a foundational model in probability theory and statistics, valued for its analytical tractability and flexibility in fitting positively skewed data.